Matching markets with middlemen under transferable utility

TL;DR

This paper analyzes matching markets with middlemen, showing how to find optimal matchings, proving core non-emptiness, and establishing the relationship between core and competitive equilibrium payoffs.

Contribution

It introduces a framework for matching markets with middlemen, demonstrating optimal matchings, core properties, and equilibrium correspondence in such markets.

Findings

Optimal matching can be found via a two-sided assignment market.

The core of the market is always non-empty.

Core and competitive equilibrium payoffs coincide.

Abstract

This paper studies matching markets in the presence of middlemen. In our framework, a buyer-seller pair may either trade directly or use the services of a middleman; and a middleman may serve multiple buyer-seller pairs. Direct trade between a buyer and a seller is costlier than a trade mediated by a middleman. For each such market, we examine an associated cooperative game with transferable utility. First, we show that an optimal matching for a matching market with middlemen can be obtained by considering the two-sided assignment market where each buyer-seller pair is allowed to use the mediation service of the middlemen free of charge and attain the maximum surplus. Second, we prove that the core of a matching market with middlemen is always non-empty. Third, we show the existence of a buyer-optimal core allocation and a seller-optimal core allocation. In general, the core does not exhibit a middleman-optimal matching. Finally, we establish the coincidence between the core and the set of competitive equilibrium payoff vectors.